Question

A deposit earns interest at a rate of r percent compounded continuously and doubles in value in 9 years. Find r. (Round your answer to two decimal places.) g

Answer 1

The earning rate is approximately 0.08.

Explanation:

We can determine the yearly rate by means of compound interest, which is defined by:

`C(t) = C_(o)\cdot (1+r)^(t)` (Eq. 1)

Where:

`C_(o)` - Initial deposit, measured in US dollars.

`r` - Earning rate, dimensionless.

`t` - Earning periods, measured in years.

We proceed to clear the earning rate within:

`(C(t))/(C_(o)) = (1+r)^(t)`

`\log (C(t))/(C_(o)) = t\cdot \log (1+r)`

`(1)/(t)\cdot \log (C(t))/(C_(o)) = \log (1+r)`

`\log \left((C(t))/(C_(o)) \right)^{(1)/(t) } = \log (1+r)`

`\left((C(t))/(C_(o)) \right)^{(1)/(t) } = 1+r`

`r = \left((C(t))/(C_(o)) \right)^{(1)/(t) }-1`

If we know that
`C(9) = 2\cdot C_(o)` and
`t = 9`, then the earning rate is:

`r = 2^{(1)/(9) }-1`

`r \approx 0.08`

The earning rate is approximately 0.08.